How do you factor completely $a^2 - 2ab - 15b^2$ ?

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Answer 1 · 2016-04-18T06:09:31

$a^2-2ab-15b^2 = (a-5b)(a+3b)$

Since this is a homogeneous polynomial of degree $2$ , you can treat it like a quadratic in one variable.

Find a pair of factors of $15$ which differ by $2$ . The pair $5, 3$ works, hence:

$a^2-2ab-15b^2 = (a-5b)(a+3b)$

Alternatively, you can complete the square and use the difference of squares identity:

$A^2-B^2=(A-B)(A+B)$

with $A=(a-b)$ and $B=4b$ as follows:

$a^2-2ab-15b^2$

$=(a-b)^2-b^2-15b^2$

$=(a-b)^2-(4b)^2$

$=((a-b)-4b)((a-b)+4b)$

$=(a-5b)(a+3b)$

How do you factor completely a^2 - 2ab - 15b^2a^2 - 2ab - 15b^2?